| L(s) = 1 | + 2.77·2-s + 5.68·4-s − 2.29·5-s − 5.27·7-s + 10.2·8-s − 6.36·10-s − 1.67·11-s + 3.70·13-s − 14.6·14-s + 16.9·16-s + 2.24·17-s + 3.22·19-s − 13.0·20-s − 4.65·22-s + 23-s + 0.279·25-s + 10.2·26-s − 29.9·28-s + 29-s + 5.83·31-s + 26.4·32-s + 6.22·34-s + 12.1·35-s − 1.40·37-s + 8.93·38-s − 23.4·40-s + 7.14·41-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + 2.84·4-s − 1.02·5-s − 1.99·7-s + 3.60·8-s − 2.01·10-s − 0.506·11-s + 1.02·13-s − 3.90·14-s + 4.23·16-s + 0.544·17-s + 0.739·19-s − 2.91·20-s − 0.991·22-s + 0.208·23-s + 0.0559·25-s + 2.01·26-s − 5.66·28-s + 0.185·29-s + 1.04·31-s + 4.68·32-s + 1.06·34-s + 2.04·35-s − 0.231·37-s + 1.44·38-s − 3.70·40-s + 1.11·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.471080807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.471080807\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.77T + 2T^{2} \) |
| 5 | \( 1 + 2.29T + 5T^{2} \) |
| 7 | \( 1 + 5.27T + 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 - 3.22T + 19T^{2} \) |
| 31 | \( 1 - 5.83T + 31T^{2} \) |
| 37 | \( 1 + 1.40T + 37T^{2} \) |
| 41 | \( 1 - 7.14T + 41T^{2} \) |
| 43 | \( 1 - 8.82T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 5.86T + 59T^{2} \) |
| 61 | \( 1 - 4.02T + 61T^{2} \) |
| 67 | \( 1 + 4.06T + 67T^{2} \) |
| 71 | \( 1 + 8.18T + 71T^{2} \) |
| 73 | \( 1 + 1.30T + 73T^{2} \) |
| 79 | \( 1 - 9.03T + 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 8.58T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.60169489392381332675419446979, −7.18318377900730666643651641034, −6.34969970655714765392750752813, −5.96671739307184554719313290950, −5.20616468884521723501769202614, −4.17833748175603285243989478413, −3.70166032761857087576812611352, −3.13570643454235548727931650349, −2.55386795489259107811746941793, −0.911544475283816709608484356413,
0.911544475283816709608484356413, 2.55386795489259107811746941793, 3.13570643454235548727931650349, 3.70166032761857087576812611352, 4.17833748175603285243989478413, 5.20616468884521723501769202614, 5.96671739307184554719313290950, 6.34969970655714765392750752813, 7.18318377900730666643651641034, 7.60169489392381332675419446979
